Countably compact spaces with \(n\)-in-countable weak bases are metrizable
Following \textit{A. V. Arhangels'skij} [Russ. Math. Surv. 21, No.4, 115--162 (1966); translation from Usp. Mat. Nauk 21, No .4(130), 133--184 (1966; Zbl 0171.43603)], a collection \({\mathcal B}= \bigcup\{{\mathcal B}_x\mid x\in X\}\) of subsets of a topological space \(X\) is called a weak base of \(X\) if, for each \(x\in X\), \({\mathcal B}_x\) is closed under finite intersections and \(x\in\bigcap{\mathcal B}_x\), and a subset \(U\) of \(X\) is open if and only if for each \(x\in U\), there is a \(B\in{\mathcal B}_x\) such that \(x\in B\subset U\).NEWLINENEWLINE In a recent paper, the author and \textit{Z. Yun} proved some interesting results concerning Tychonoff spaces with an \(n\)-in-countable (weak) base, where, for every positive integer \(n\), a collection \({\mathcal B}\) of subsets of a set \(X\) is called \(n\)-in-countable if every \(A\subset X\) with \(|A|=n\) is contained in only finitely many elements of \({\mathcal B}\) [Topology Appl. 159, No. 17, 3545--3549 (2012; Zbl 1256.54051)]. Some of the problems that were left open in this paper have been solved in the reviewed paper.NEWLINENEWLINE The main result is mentioned in the title. It answers a question raised by \textit{H. R. Bennett} and \textit{H. W. Martin} [Port. Math. 36, 247--250 (1977; Zbl 0448.54032)]. Its interesting proof uses the non-trivial fact, first proven by \textit{V. V. Tkachuk} [Topology Appl. 156, 840--846 (2009; Zbl 1165.54009)], that every monotonically monolithic Tychonoff space is a D-space. Every Tychonoff space with an \(n\)-in-countable weak base for some \(n\) is weakly monotonically monolithic, and every Tychonoff space with an \(n\)-in-countable base for some \(n\) is monotonically monolithic. There are several more interesting results concerning Tychonoff spaces with an \(n\)-in-countable base, which are omitted here.NEWLINENEWLINE Finally, two questions of \textit{V. V. Tkachuk} [Topology Appl. 159, No. 6, 1529--1535 (2012; Zbl 1245.54022)] are answered in the negative.
scientific article; zbMATH DE number 6334514